Quadratic-polytrope disk with constant aspect ratio ($\Sigma =\frac{GM}{K}[\mathrm{arsinh}ϵ-ϵ/\sqrt{1+{ϵ}^2}]$)

For $p=K{\rho }^2$, a point-mass gravitational potential and a zero-density free surface at $z=\pm ϵr$, the barotropic cylindrical rotation theorem gives ${\Omega }^2=GM/(r^3\sqrt{1+{ϵ}^2})$. Vertical balance gives $\rho =GM[1/\sqrt{r^2+z^2}-1/(r\sqrt{1+{ϵ}^2})]/(2K)$. Integrating over the finite height produces the displayed constant surface density. This is a local radial equilibrium family; infinite radial extent would have infinite mass.